Sharp, giả giá thứ i chiều lớn hơn 0 của M là giá suy rộng thứ i của M được giới thiệu bởi Lê Thanh Nhàn, Nguyễn Thị Kiều Nga và Phạm Hữu Khánh (2014).. Các tập này là các công cụ hữu [r]
Trang 138 http://jst.tnu.edu.vn; Email: jst@tnu.edu.vn
ON SOME PROPERTIES OF THE PSEUDO SUPPORT
IN DIMENSION MORE THAN S
Nguyen Thi Anh Hang 1* , Nguyen Thi Yen 2
1 TNU - University of Education,
2 College of Artillery of Officer’s Training
ABSTRACT
Let (R, m) be a Noetherian local ring and M be a finitely generated Rmodule Let s and i be integers such that i ≥ 0 and s ≥ −1 The i-th pseudo support in dimension more than s of M, denoted by PsuppiR(M) >s, is defined by PsuppiR(M) >s = {p ∈ Spec R | N-dimRp Hpi −Rdimp R/ p(Mp) > s} The i-th pseudo support in dimension more than −1 of M is the i-th pseudo support
of M introduced by Markus Brodmann and R Y Sharp, the i-th pseudo support in dimension more than 0 of M is the i-th length support of M introduced by Le Thanh Nhan, Nguyen Thi Kieu Nga
and Pham Huu Khanh (2014) They are useful tools in studying Cohen-Macaulay loci In this
paper, we are going to present some properties of the i-th pseudo support in dimension more than s
of M
Key words: the i-th pseudo support; catenary; universally catenary; attached primes; formal fibres.
Received: 12/5/2020; Revised: 22/8/2020; Published: 27/8/2020
MỘT SỐ TÍNH CHẤT CỦA TẬP GIẢ GIÁ CHIỀU LỚN HƠN S
Nguyễn Thị Ánh Hằng 1* , Nguyễn Thị Yến 2
1 Trường Đại học Sư phạm – ĐH Thái Nguyên,
2 Trường Sĩ quan pháo binh
TÓM TẮT
Cho (R, m) là một vành địa phương Noether và M là một R-môđun hữu hạn sinh Cho i ≥ 0 và s ≥ −1 là các số nguyên Giả giá thứ i chiều lớn hơn s của M, được ký hiệu bởi PsuppiR(M) >s, và được định nghĩa là tập PsuppiR(M) >s = { p ∈ Spec R | N-dimRp Hpi −Rdimp R/ p(Mp) > s} Giả giá
thứ i chiều lớn hơn −1 của M là giả giá thứ i của M được giới thiệu bởi Markus Brodmann và R
Y Sharp, giả giá thứ i chiều lớn hơn 0 của M là giá suy rộng thứ i của M được giới thiệu bởi Lê
Thanh Nhàn, Nguyễn Thị Kiều Nga và Phạm Hữu Khánh (2014) Các tập này là các công cụ hữu ích trong nghiên cứu các quỹ tích liên quan đến tính Cohen-Macaulay của môđun Trong bài báo
này, chúng tôi trình bày một số tính chất của tập giả giá thứ i chiều lớn hơn s của M
Từ khóa: Tập giả giá thứ i; tính catenary; tính catenary phổ dụng; tập các iđêan nguyên tố gắn kết; thớ hình thức
Ngày nhận bài: 12/5/2020; Ngày hoàn thiện: 22/8/2020; Ngày đăng: 27/8/2020
* Corresponding author Email: hangnta@tnue.edu.vn
https://doi.org/10.34238/tnu-jst.3119
Trang 21 Introduction
Throughout this paper, let (R, m) be a
Noetherian local ring and M a finitely
gen-erated R-module with dim M = d Let
i ≥ 0 and s ≥ −1 be integers
Follow-ing M Brodmann and R Y Sharp ([1]),
the i-th pseudo support of M, denoted by
{p ∈ Spec R | Hpi−dim R/ pRp (Mp) 6= 0}
The pseudo supports of M play an
impor-tant role in studying the dimension and
multiplicity of local cohomologies with
re-spect to the maximal ideal Pseudo
sup-ports are also very useful in describing the
non-Cohen-Macaulay locus nCM(M ) of M
(see [2])
The notion of the length support was
in-troduced by Nhan, Nga and Khanh in [3]
The ith length support of M , denoted by
{p ∈ Spec(R)|`Rp(Hpi−dim R/ pR
The length supports are effective in
in-vestigating the non-generalized
Cohen-Macaulay locus nGCM(M ) of M (see [3])
In [4], L P Thao introduced the notion
of the i-th pseudo support in dimension
more than s of M and described the
non-Cohen-Macaulay locus in dimension more
than s via the i-th pseudo support in
di-mension more than s of M
Definition 1.1 The i-th pseudo support
in dimension more than s of M, denoted
N-dimR p Hpi−dim R/ pRp (Mp) > s}
Note that if s = −1 then the i-th pseudo
support in dimension more than −1 of M
is the i-th pseudo support of M If s = 0,
then the i-th pseudo support in dimension
more than 0 of M is i-th length support of
M
In this paper, the i-th pseudo support in dimension more than s of M under comple-tion, localizacomple-tion, and other its properties are investigated
2 Main results For each ideal I of R, denote by Var(I) the set of all prime ideals of R containing I First, we recall the property (*) for an Ar-tinian R-module A, which was considered firstly by N T Cuong and L T Nhan (see [5])
If R is complete with respect to m-adic topology, it follows by Matlis duality that the property (*) is satisfied for all Artinian R-modules A When R is universally cate-nary and all its formal fibres are
for any integer i, cf [6]
Lemma 2.1 The following statements are true:
satisfies the property (*) if and only if
univer-sally catenary and all its formal fibres are
property (*) for all i
The following lemma proved that the property (*) of local cohomology is un-changed under localization (see [7])
Lemma 2.2 Let i ≥ 0 be an integer
the following statements are equivalent:
(ii) Hpi−dim R/ pR
(*) for all p ∈ Supp(M )
The property (*) of an Artinian module
A is a sufficient condition It means that the
Trang 3Krull dimension of A is equal to the
Noe-therian dimension of A (see [5], Proposition
4.6)
Lemma 2.3 Let A be an Artinian module
If A satisfies the property (*) then
catenary and all its formal fibres are
Cohen-Macaulay then by the Lemma 2.1,
Lemma 2.2, and Lemma 2, we have
dimRRp/ AnnRpHpi−dim R/ pR
(ii) PsuppiR(M )>s ⊆ Psuppi
Now, we collect some properties of the
pseudo support in the following lemma
These are known and can be found in [1],
[2] and [7] For each subset T of Spec(R)
Lemma 2.5 (i) If R is catenary then
The following lemma is a key one of the
paper
Lemma 2.6 If R is universally
cate-nary and all its formal fibres are
closed under specialization
dim R/ p) − ht p / q, which is equal to (i −
dim R/ p) − (dim R/ q − dim R/ p) Hence
So that Hi−dim(R/ p)−dim(Rp / q R p )
Psuppi−dim(R/ p)R
dim Rp/ AnnR pHpi−dim(R/ p)Rp (Mp) > s
So, p ∈ PsuppiR(M )>s
Since R is universally catenary and all
dimRRp/ AnnRpHpi−dim R/ pRp (Mp) > s Then there exists q Rp ∈ AttR pHpi−dim(R/ p)Rp (Mp)
p, q ∈ PsuppiR(M ), ht(pq) ≤ s}
we have ht(q/θ) > s Hence ht(p/θ) > s
Corollary 2.7 If R is universally cate-nary and all its formal fibers are
of one of the following sets
s}
s}
s}
Trang 4Let bR and cM be the m-adic completion
of R and M , respectively The following
re-sult gives a new property of the i-th pseudo
support in dimension more than s of M
un-der completion
Proposition 2.8 Assume that R is
uni-versally catenary and all its formal fibers
b
R(M ), Hpi−dim(R/ p)R
b
b
satisfies the Going up property It means
such that ht(P/Q) ≥ ht(p / q) > s and
Q∩R = q Since R is catenary, dim(R/ q) ≥
≥ dim(R/ p) + ht(p / q)
= dim(R/ q)
b
b
The following example show that we can
have the strict inclusion in Proposition 2.8
Example 2.9 Let t > 0 be an inte-ger Then there exists a Noetherian lo-cal ring (R, m) such that R is a quotion
P(bRP / p bRP) so
b
b
R(bR / p bR) since
b
specializa-tion Since ht(P/Q > 0), by the Lemma
b
R(bR / p bR) and [7], Proposition 3.2 It deduces that
Finally, we study the i-th pseudo sup-port in dimension more than s of M under localization The results is presented in the following theorem
Theorem 2.10 Assume that R is univer-sally catenary and all its formal fibers are
Psuppi−dim(R/ p)Rp (Mp)>s is the set
q Rp | q ∈ PsuppiR(M )>s, q ⊆ p Proof Let q Rp ∈ Psuppi−dim(R/ p)Rp (Mp)>s
Corollary 2.7
qRp ⊇ q1Rp∈ min AttRp Hpi−dim R/ pRp (Mp) Since R is universally catenary and all its formal fibers are Cohen-Macaulay,
by [7] and the Lemma 2.2, we have
Trang 5q ∈ PsuppiR(M ), q ⊆ p and
min AttRp Hpi−dim R/ pRp (Mp) is equal to the
Psuppi−dim(R/ p)R
p∈ PsuppiR(M )>s
2.1 This implies
qRq These follow that HqRi−dim(R/ q)q (Mq) ∼=
min AttRqHqRi−dim(R/ q)q (Mq) It deduces that
3 Conclusion Let (R, m) be a Noetherian local ring and
M be a finitely generated R-module Let
s and i be integers such that i ≥ 0 and
s ≥ −1 In this paper, we have present some
new properties of the i-th pseudo support
in dimension more than s of M under com-pletion and localization
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T Nguyen, "On pseudo supports and non-Cohen-Macaulay locus of finitely generated modules," Journal of Algebra, vol 323, pp 3029-3038, 2010
[3] N T Le, N K T Nguyen and
K H Pham, "Non Cohen-Macaulay locus and non generalized Cohen-Macaulay lo-cus,"Communications in Algebra, vol 42,
pp 4414-4425, 2014
[4] T P Luu, "Non Cohen-Macaulay in dimension > s locus," TNU Journal of Sci-ence and Technology, vol 192, no 16, pp 23-28, 2018
[5] C T Nguyen, "On the Noetherian dimension of Artinian modules," Vietnam Journal of Mathematics, vol 30, no 2, pp 121-130, 2002
[6] N T Le and A N Tran, ”On the un-mixedness and the universal catenaricity of local rings and local cohomology modules,” Journal of Algebra, vol 321, pp 303-311,
2009 [7] A N Tran, "On the attached primes and shifted localization principle for lo-cal cohomology modules ," Algebra Collo-quium, vol 20, pp 671-680, 2013